diff --git a/ChangeLog.md b/ChangeLog.md --- a/ChangeLog.md +++ b/ChangeLog.md @@ -1,3 +1,9 @@ +# 0.0.4 (2014-03-27) + +* Make the description more precise. +* Add continuedFractionApproximate for rational partial evaluations of + continued fractions. + # 0.0.3 (2014-03-26) * Add a more verbose description of the library. diff --git a/README.md b/README.md --- a/README.md +++ b/README.md @@ -2,8 +2,9 @@ [![Build Status](https://travis-ci.org/ion1/quadratic-irrational.svg)](https://travis-ci.org/ion1/quadratic-irrational) -An implementation of [quadratic irrationals][qi] with support for conversion -from and to [periodic continued fractions][pcf]. +A library for exact computation with [quadratic irrationals][qi] with support +for exact conversion from and to [(potentially periodic) simple continued +fractions][pcf]. [qi]: http://en.wikipedia.org/wiki/Quadratic_irrational [pcf]: http://en.wikipedia.org/wiki/Periodic_continued_fraction @@ -24,19 +25,19 @@ * `(1 + √5)/2` ([the golden ratio][gr]), -* solutions to some quadratic equations – the [quadratic formula][qf] has a - familiar shape. +* solutions to quadratic equations with rational constants – the [quadratic + formula][qf] has a familiar shape. [gr]: http://en.wikipedia.org/wiki/Golden_ratio [qf]: http://en.wikipedia.org/wiki/Quadratic_formula -A continued fraction is a number that can be expressed in the form +A simple continued fraction is a number in the form ``` a + 1/(b + 1/(c + 1/(d + 1/(e + …)))) ``` -alternatively expressed using the notation +or alternatively written as ``` [a; b, c, d, e, …] @@ -44,8 +45,8 @@ where `a` is an integer and `b`, `c`, `d`, `e`, … are positive integers. -Every finite continued fraction represents a rational number and every -infinite, periodic continued fraction represents a quadratic irrational. +Every finite SCF represents a rational number and every infinite, periodic SCF +represents a quadratic irrational. ``` 3.5 = [3; 2] diff --git a/quadratic-irrational.cabal b/quadratic-irrational.cabal --- a/quadratic-irrational.cabal +++ b/quadratic-irrational.cabal @@ -1,6 +1,6 @@ name: quadratic-irrational category: Math, Algorithms, Data -version: 0.0.3 +version: 0.0.4 license: MIT license-file: LICENSE author: Johan Kiviniemi @@ -11,10 +11,10 @@ copyright: Copyright © 2014 Johan Kiviniemi synopsis: An implementation of quadratic irrationals description: - An implementation of + A library for exact computation with - with support for conversion from and to - . + with support for exact conversion from and to + . . A quadratic irrational is a number that can be expressed in the form . @@ -31,22 +31,22 @@ * @(1 + √5)\/2@ (), . - * solutions to some quadratic equations – the + * solutions to quadratic equations with rational constants – the has a familiar shape. . - A continued fraction is a number that can be expressed in the form + A simple continued fraction is a number expressed in the form . > a + 1/(b + 1/(c + 1/(d + 1/(e + …)))) . - alternatively expressed using the notation + or alternatively written as . > [a; b, c, d, e, …] . where @a@ is an integer and @b@, @c@, @d@, @e@, … are positive integers. . - Every finite continued fraction represents a rational number and every - infinite, periodic continued fraction represents a quadratic irrational. + Every finite SCF represents a rational number and every infinite, periodic + SCF represents a quadratic irrational. . > 3.5 = [3; 2] > (1+√5)/2 = [1; 1, 1, 1, …] diff --git a/src/Numeric/QuadraticIrrational.hs b/src/Numeric/QuadraticIrrational.hs --- a/src/Numeric/QuadraticIrrational.hs +++ b/src/Numeric/QuadraticIrrational.hs @@ -9,10 +9,10 @@ -- Stability : provisional -- Portability : ViewPatterns -- --- An implementation of +-- A library for exact computation with -- --- with support for conversion from and to --- . +-- with support for exact conversion from and to +-- . -- -- A quadratic irrational is a number that can be expressed in the form -- @@ -29,22 +29,22 @@ -- * @(1 + √5)\/2@ -- (), -- --- * solutions to some quadratic equations – the +-- * solutions to quadratic equations with rational constants – the -- has a -- familiar shape. -- --- A continued fraction is a number that can be expressed in the form +-- A simple continued fraction is a number expressed in the form -- -- > a + 1/(b + 1/(c + 1/(d + 1/(e + …)))) -- --- alternatively expressed using the notation +-- or alternatively written as -- -- > [a; b, c, d, e, …] -- -- where @a@ is an integer and @b@, @c@, @d@, @e@, … are positive integers. -- --- Every finite continued fraction represents a rational number and every --- infinite, periodic continued fraction represents a quadratic irrational. +-- Every finite SCF represents a rational number and every infinite, periodic +-- SCF represents a quadratic irrational. -- -- > 3.5 = [3; 2] -- > (1+√5)/2 = [1; 1, 1, 1, …] @@ -64,11 +64,13 @@ , qiFloor , -- * Continued fractions continuedFractionToQI, qiToContinuedFraction + , continuedFractionApproximate , module Numeric.QuadraticIrrational.CyclicList ) where import Control.Applicative import Control.Monad.State +import qualified Data.Foldable as F import Data.List import Data.Maybe import Data.Ratio @@ -591,7 +593,7 @@ ~(b2cLow, b2cHigh) = iSqrtBounds (b*b * c) --- | Convert a (possibly periodic) continued fraction to a 'QI'. +-- | Convert a (possibly periodic) simple continued fraction to a 'QI'. -- -- @[2; 2] = 2 + 1\/2 = 5\/2@. -- @@ -647,7 +649,7 @@ pos = positive "continuedFractionToQI" --- | Convert a 'QI' into a (possibly periodic) continued fraction. +-- | Convert a 'QI' into a (possibly periodic) simple continued fraction. -- -- @5\/2 = 2 + 1\/2 = [2; 2]@. -- @@ -694,6 +696,20 @@ go (Just n) = (unQI n, i) : go (qiRecip (qiSubI n i)) where i = qiFloor n go Nothing = [] + +-- | Compute a rational partial evaluation of a simple continued fraction. +-- +-- Rational approximations that converge toward φ: +-- +-- >>> [ continuedFractionApproximate n (1, repeat 1) | n <- [0,3..18] ] +-- [1 % 1,5 % 3,21 % 13,89 % 55,377 % 233,1597 % 987,6765 % 4181] +continuedFractionApproximate :: F.Foldable f + => Int -> (Integer, f Integer) -> Rational +continuedFractionApproximate n (i0, F.toList -> is) = + fromInteger i0 + + foldr (\(pos -> i) r -> recip (fromInteger i + r)) 0 (take n is) + where + pos = positive "continuedFractionApproximate" iSqrtBounds :: Integer -> (Integer, Integer) iSqrtBounds n = (low, high) diff --git a/tests/QuadraticIrrational.hs b/tests/QuadraticIrrational.hs --- a/tests/QuadraticIrrational.hs +++ b/tests/QuadraticIrrational.hs @@ -156,6 +156,11 @@ -- Limit the length of the periodic part for speed. in (len <= 100) ==> approxEq' (qiToFloat n) (qiToFloat (continuedFractionToQI cf)) + + , testProperty "continuedFractionApproximate" $ \n -> + let cf = qiToContinuedFraction n + n' = continuedFractionApproximate 20 cf + in approxEq' (qiToFloat n) (fromRational n') ] ]